Algebraic geometry can be thought of as the study of generalizations of the Fundamental Theorem of Algebra. Precisely, it is the study of the zero sets of polynomials. So the FTA is just the restriction to the case of a single polynomial in one variable.

There are many algebraic structures which may be given to sets in order to solve problems and bring out some extraordinary properties. Among these are groups, rings, fields, vector fields, modules, algebras, categories, and of course all the finer structures (subrings, domains, and the like). As one might guess, abstract algebra is very heavy in definitions, but each of these structures has its own place, and each is widely applicable in many fields of mathematics.

The main goal in abstract algebra is extending the operations and properties we take for granted on sets we're used to working with (like integers, reals, complex numbers, etc.) to arbitrary sets. This requires precise definitions and requirements on the structure of the set in order to ensure the desired properties are present.